On Prüfer-Like Properties of Leavitt Path Algebras

Songül Esin, Müge Kanuni, Ayten Koç, Katherine Radler

Published 2018-08-30Version 1

Dedekind domains and Pr\"{u}fer domains admit several characterizations by means of the properties of their ideal lattices. Interestingly, a Leavitt path algebra $L$, in spite of being non-commutative possessing plenty of zero divisors, seem to have its ideal lattice possess these characterizing properties of these special domains. In [6] it was shown that the ideals of $L$ satisfy the distributive law and the Chinese Remainder Theorem holds for these ideals. In this paper we show two more characterizing properties of a Pr\"{u}fer domain hold in a Leavitt path algebra and we also show that one of the properties (the cancellation property) of a Pr\"{u}fer domain does not hold by giving a counterexample. It is also investigated that under what conditions the cancellative property holds among the ideals of $L$.